## proving that a problem about grammar is unsolvable

I came across this problem in the Martin D. Davis book about computability and on page 192, exercise 1(1):

show that there is no algorithm to determine of a given grammar G whether L(G) contains at least one word with exactly three symbols.


I understand that there is no algorithm to determine whether a given word w belongs to a given language L(G) generated by a given grammar G, and there is no algorithm to determine whether two given L(G1) L(G2) intersect, but the proof constructed a special counter example which might not be suitable for this problem.

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